Murasugi sum of k –open books

A k–open book of a closed n–manifold X is a way to express X as a fiber bundle π: X\ B→ Sk over the k–sphere Sk in the complement of a (n- k- 1) –dimensional submanifold B of X. One can associate an abstract k–open book to a given k–open book of a closed manifold. Given an abstract k–open book of a closed manifold X and an abstract k–open book of a closed manifold X′ , we define the notion of their Murasugi sum and show that the closed manifold associated to the Murasugi sum is the connected sum of X and X′ .